Algebraic Curves/Fall 2015
نویسنده
چکیده
(Algebra) It’s a complex projective manifold C of dimension one. As such, it has a field of rational functions K(C) of transcendence degree one over the field C of scalars. The curve can be recovered from the field as the set of discrete valuation rings of K/C, which has the “natural” structure of a projective manifold. There is a canonical line bundle ωC , whose degree captures the genus via deg(ωC) = 2g − 2. Alternatively, the space of sections of this line bundle has dimension g, and the “algebraic” Euler characteristic of C is 1− g.
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